A Finsler Geodesic Flow on T^2 With Positive Metric Entropy
aa r X i v : . [ m a t h . DG ] F e b A Finsler Geodesic Flow On T With Positive MetricEntropy
Stefan Klempnauer ∗ Faculty of Mathematics, Ruhr-University BochumFebruary 8, 2021
Abstract
We use a theorem of P. Berger and D. Turaev to construct an example of a Finslergeodesic flow on the 2-torus with a transverse section, such that its Poincar´e returnmap has positive metric entropy. The Finsler metric generating the flow can bechosen to be arbitrarily C ∞ -close to a flat metric. The author thanks the SFB CRC/TRR 191
Symplectic Structures in Geometry, Algebraand Dynamics of the DFG and the Ruhr-University Bochum for the funding of hisresearch.
We give an example of a Finsler geodesic flow on the 2-torus that exhibits aspects ofboth chaotic and integrable dynamics. The flow exhibits integrable behaviour in thesense that a a large region of the unit tangent bundle is foliated by invariant tori, oneach of which the geodesic flow is a linear flow. It is chaotic in the sense that its returnmap of a certain transverse section contains a stochastic island, i.e. a region of positivemetric entropy.In section 2 we use a result of P. Berger and D. Turaev to obtain a perturbation of thestandard shear map of the cylinder with positive metric entropy. In section 3 this mapis embedded as a return map of the geodesic flow on the unit tangent bundle ST . ˆ f with positive metric entropy Let (
M, ω ) be a surface with a smooth area form. Let f : M → M be a diffeomorphism.The maximal Lyapunov exponent of x ∈ M is given by λ ( x ) = lim sup n →∞ n log k Df n ( x ) k ∗ [email protected] f preserves ω then we have that the metric entropy of f is given by h ω ( f ) := Z M λ ( x ) dµ ω From [1] we have the following theorem
Theorem 2.1. (Berger, Turaev ’19)
Let ( M, ω ) be a surface with a smooth area form. If f : M → M is a smooth area-preserving diffeomorphism with a non-hyperbolic periodicpoint, then there is an arbitrarily C ∞ -small perturbation of f , such that the perturbedmap ˆ f is a smooth, area-preserving diffeomorphism and has positive metric entropy h ω ( ˆ f ) > . Remark 2.2.
The theorem is proved in [1] by perturbing the diffeomorphism f locallyalong the orbit of the periodic point, such that one obtains an invariant domain I ⊂ M of positive measure with positive maximal Lyapunov exponent at every point in I . Sincethe perturbation is local the perturbed map ˆ f agrees with f away from the orbit of theperiodic point. The boundary of I consists of finitely many C -embedded circles that liein the union of the stable and unstable manifolds of a set of hyperbolic periodic points.In the spirit of [1] we call such a set I a stochastic island . Let Z = S × R be the cylinder equipped with the standard symplectic form dx ∧ dy .The shear map f : Z → Z with ( x, y ) ( x + y mod 1 , y ) is the simplest exampleof a twist map of the cylinder. Note that f has a non-hyperbolic fixed point at (0 , C ∞ diffeomorphism ˆ f : Z → Z with a stochastic island I , which is arbitrarily closeto f in C ∞ . The twist of f is uniformly equal to 1. Thus, if ˆ f is C -close enough to f it has bounded twist away from zero. Note that if ˆ f is C -close enough to f thenKAM-theory guarantees that ˆ f possesses invariant essential circles and consequently haszero flux (see [2]). Consequently, ˆ f is a twist map of the cylinder if it is close enough to f in C r for r > K >
0, such that ˆ f ( x, y ) = f ( x, y ) for every | y | > K . ˆ f into the geodesic flow We identify the tangent bundle
T T with T × R . A Finsler metric on T is a map F : T T → [0 , ∞ )with the following properties1. (Regularity) F is C ∞ on T T −
02. (Positive homogeneity) F ( x, λy ) = λ · F ( x, y ) for λ >
03. (Strong convexity) The hessian( g ij ( x, v )) = (cid:18) ∂ v i v j F ( x, v ) (cid:19) is positive-definite for every ( x, v ) ∈ T T − reversible if F ( x, v ) = F ( x, − v ) for every ( x, v ) ∈ T T . The unit tangent bundle ST ∼ = T × S is given by ST = F − ( { } ).The geodesic flow φ t : ST → ST is the restriction of the Euler-Lagrange flow of theLagrangian L F , with L F = 12 F to the unit tangent bundle. A geodesic we call either a trajectory of the Euler-Lagrangeflow, or its projection to T , i.e. a geodesic is a curve t c ( t ) ⊂ T satisfying theEuler-Lagrange equation ∂ x L F ( c, ˙ c ) − ∂ t ( ∂ v L F ( c, ˙ c )) = 0To embed the map ˆ f into the geodesic flow we use a theorem of Moser [3] to express ˆ f as the time-1 map of a strictly convex, time-periodic Hamiltonian on S . Theorem 3.1. (J. Moser ’86)
Given a C ∞ twist map f : Z → Z with f ( x, y ) =( x + c · y, y ) for large | y | , there exists a strictly convex, time-periodic Hamiltonian H on S , such that the time-1 map ψ , H : Z → Z agrees with f . Remark 3.2.
Moser’s original theorem is formulated for twist maps f on the closedannulus A = S × [0 , . In the proof in [3] the map f is extended to a twist map on thecylinder Z with f ( x, y ) = ( x + c · y, y ) for | y | > D for a positive constant D . Thereexist constants D + , D − ∈ R , such that the Hamiltonian H is equal to y + D ± for largevalues of | y | , depending on whether y > D or y < − D . Let H be the Hamiltonian obtained from theorem 3.1 generating the previously con-structed twist map ˆ f . We lift H to obtain a e · Z -periodic Hamiltonian H on R . TheLegendre transformation L t : R → R is a global diffeomorphism and agrees with theidentity for large values of | y | . Thus, we obtain an associated Lagrangian ˆ L : S × R → R with ˆ L ( t, x, y ) = 12 y − D ± for | y | > D Observe that the time-dependent Euler-Lagrange flow of ˆ L is complete (i.e. the Euler-Lagrange solutions exist for all times) because the sets { y = const. } are invariant forlarge | y | . To embed ˆ L into a Finsler metric we need to perturb it for large values of | y | .Let h + , h − : R → R be smooth functions with h + ( y ) = (cid:26) y − D + if y < D + 1 p A + By if y > D + 2and h − ( y ) = (cid:26) p A + By if y < − D − y − D − if y > − D − A, B > h + , h − with h ′′± >
0. We define a Lagrangian L via L ( t, x, y ) = h − ( y ) if y < − D ˆ L ( t, x, y ) if − D ≤ y ≤ Dh + ( y ) if y > D Observe that the time-dependent flow of ˆ L is the same as the time-dependent flow of L since ˆ L and L only differ where they are both only dependent on y . Let F be a Finslermetric on T given by F ( t, x, v , v ) = q Av + Bv L is chosen in such a way that L ( t, x, y ) = F ( t, x, , y ) for large valuesof | y | . We define a map F on T via F ( t, x, v , v ) = (cid:26) v · L ( t, x, v v ) if v > F ( t, x, v , v ) if v ≤ Remark 3.3.
From the proof of Moser’s theorem 3.1 it follows that if ˆ f is chosen C ∞ -close to the shear map then the obtained Hamiltonian will be C ∞ -close to H = y . Consequently, the above obtained Lagrangian L can be chosen to be C ∞ -close to afunction h only dependent on y with h ( y ) = (cid:26) y if | y | < D + 1 p A + By if | y | > D + 2 Thus, for any compact subset K of T T we can find a sequence ˆ f i of twist maps con-verging to f and do the above construction, such that the resulting Finsler metrics F i become arbitrarily C ∞ -close to the flat metric ¯ F ( t, x, v , v ) = (cid:26) v · h ( v v ) if v > F ( t, x, v , v ) if v ≤ on the set K . The following two propositions are due to J.P. Schr¨oder [4]. We include their proofs forcompleteness.
Proposition 3.4. F defines a C ∞ Finsler metric on T .Proof. Regularity and Positive homogeneity follow directly from the definition of F .We check strict convexity in each fiber. Let ( t, x ) ∈ T be fixed and define f : R > × R → R via f ( v , v ) := v · l ( v /v )where l ( y ) := L ( t, x, y ) for every y ∈ R . We compute the derivatives ∂ f ( u , u ) = l (cid:18) u u (cid:19) − u u · l ′ (cid:18) u u (cid:19) ∂ f ( u , u ) = l ′ (cid:18) u u (cid:19) second derivatives ∂ f ( u , u ) = u u · l ′′ (cid:18) u u (cid:19) ∂ f ( u , u ) = ∂ f ( u , u ) = − u u · l ′′ (cid:18) u u (cid:19) ∂ f ( u , u ) = 1 u · l ′′ (cid:18) u u (cid:19) Consequently, we have for u = ( u , u ) ∈ R > × R and v = ( v , v ) ∈ R h v, Hessf ( u ) v i = (cid:18) v · u u − v (cid:19) · l ′′ ( u /u ) u
4o see that F is strictly convex we have to check that fiberwise the Hessian of L F = F is positive definite. For ( t, x ) ∈ T fixed let L F : T ( t,x ) T → R be given by L F ( u ) = F ( t, x, u ) . For u ∈ R > × R we then have L F ( u ) = 12 ( u · l ( u /u )) We compute partial derivatives ∂ L F ( u ) = f ( u ) · ∂ f ( u ) ∂ L F ( u ) = f ( u ) · ∂ f ( u ) ∂ L F ( u ) = ( ∂ f ( u )) + f ( u ) · ∂ f ( u ) ∂ L F ( u ) = ∂ f ( u ) · ∂ f ( u ) + f ( u ) · ∂ f ( u ) ∂ L F ( u ) = ( ∂ f ( u )) + f ( u ) · ∂ f ( u )from this we get that HessL F ( u ) = A + f ( u ) · Hessf ( u )where A = (cid:18) ( ∂ f ( u )) ∂ f ( u ) · ∂ f ( u ) ∂ f ( u ) · ∂ f ( u ) ( ∂ f ( u )) (cid:19) For v ∈ R we compute v T Av = v ( ∂ f ( u )) + 2 v v ∂ f ( u ) · ∂ f ( u ) + v ( ∂ f ( u )) = ( v ∂ f ( u ) + v ∂ f ( u )) = ( Df ( u ) v ) Hence we have v T HessL F ( u ) v = v T ( A + f ( u ) Hessf ( u )) v = v T Av + f ( u ) v T Hessf ( u ) v = ( Df ( u ) v ) | {z } a + f ( u ) (cid:18) v u u − v (cid:19) l ′′ ( u /u ) u | {z } b Observe that a and b are each ≥
0. Assume now, that b = 0. Since f, l ′′ and u are > v u u − v = 0. From this it follows that v = λ · u are linearly dependent.In this case we have Df ( u ) v = λ · Df ( u ) u = f ( v ) >
0. Hence the Hessian
HessL F ( x, u )is positive definite for u ∈ R > × R . It is also positive definite for u ∈ R ≤ × R − { } since L coincides there with the Finsler metric F . Proposition 3.5.
Let θ : R → R be a smooth function and let γ : R → R be the curvegiven by γ ( t ) = ( t, θ ( t )) Then γ is a reparametrization of a lifted F -geodesic if and only if θ is an Euler-Lagrangesolution of L (seen as a Lagrangian lifted to R ). roof. Observe that we have the following relation between the Lagrangian action A L and the Finsler length l F . A L ( θ | [ a,b ] ) = Z ba L ( t, θ ( t ) , θ ′ ( t )) dt = Z ba F ( t, θ ( t ) , , θ ′ ( t )) dt = Z ba F ( γ ( t ) , ˙ γ ( t )) dt = l F ( γ | [ a,b ] )Assume now that γ : [ a, b ] → R is a reparametrization of an F -geodesic, i.e. ∂ s =0 l F ( γ s ) =0 for any proper variation of γ . Let θ s : [ a, b ] → R be a proper variation of θ . From θ s we construct a proper variation γ s of γ via γ s ( t ) = ( t, θ s ( t ))Then we have A L ( θ s ) = l F ( γ s ) for every s , and hence we have ∂ s | s =0 A L ( θ s ) = ∂ s | s =0 l F ( γ s ) = 0 . This proves one direction. To prove the other direction assume now that θ : [ a, b ] → R is critical with respect to the Lagrangian action and let X : [ a, b ] → R be a vector fieldalong γ with X ( a ) = X ( b ) = 0. Since ˙ γ ( t ) = (1 , θ ′ ( t )) the pair of vectors { ˙ γ ( t ) , e } always forms a basis of R . Thus, we can rewrite the vector field X as X ( t ) = λ ( t ) ˙ γ ( t ) | {z } A ( t ) + µ ( t ) e | {z } B ( t ) for functions λ, µ with λ ( a ) = λ ( b ) = µ ( a ) = µ ( b ) = 0. Let γ s be a proper variation of γ corresponding to the variational vector field X and let β s be a proper variation of γ corresponding to B , i.e. ∂ s | s =0 γ s ( t ) = X ( t ) and ∂ s | s =0 β s ( t ) = B ( t )Observe that for small | s | the curve η s : t γ ( t + sλ ( t )) is a reparametrization of γ andhence has length independent of s . Thus0 = ∂ s | s =0 l F ( η s )= ∂ s | s =0 Z ba F ( η s ( t ) , ∂ t η s ( t )) dt = Z ba ∂ s | s =0 F ( η s ( t ) , ∂ t η s ( t )) dt = Z ba ∂ F ( η ( t ) , ∂ t η ( t )) ∂ s | s =0 η s ( t ) + ∂ F ( η ( t ) , ∂ t η ( t )) ∂ s | s =0 ∂ t η s ( t ) dt = Z ba ∂ F ( γ ( t ) , ∂ t γ ( t )) ∂ s | s =0 η s ( t ) + ∂ F ( γ ( t ) , ∂ t γ ( t )) ∂ t ∂ s | s =0 η s ( t ) dt = Z ba ∂ F ( γ ( t ) , ∂ t γ ( t )) A ( t ) + ∂ F ( γ ( t ) , ∂ t γ ( t )) ˙ A ( t ) dt ∂ s | s =0 l F ( γ s ) = ∂ s | s =0 Z ba F ( γ s ( t ) , ˙ γ s ( t )) dt = Z ba ∂ F ( γ ( t ) , ˙ γ ( t )) ∂ s | s =0 γ s ( t ) + ∂ F ( γ ( t ) , ˙ γ ( t )) ∂ s | s =0 ∂ t γ s ( t )= Z ba ∂ F ( γ ( t ) , ˙ γ ( t )) ∂ s | s =0 γ s ( t ) + ∂ F ( γ ( t ) , ˙ γ ( t )) ∂ t ∂ s | s =0 γ s ( t )= Z ba ∂ F ( γ ( t ) , ˙ γ ( t )) X ( t ) + ∂ F ( γ ( t ) , ˙ γ ( t )) ˙ X ( t )= Z ba ∂ F ( γ ( t ) , ˙ γ ( t ))( A ( t ) + B ( t )) + ∂ F ( γ ( t ) , ˙ γ ( t ))( ˙ A ( t ) + ˙ B ( t ))= Z ba ∂ F ( γ ( t ) , ˙ γ ( t )) B ( t ) + ∂ F ( γ ( t ) , ˙ γ ( t )) ˙ B ( t )= ∂ s | s =0 l F ( β s )Consequently, the curve γ is critical with respect to the Finsler length if ∂ s | s =0 l F ( β s ) = 0for every proper variation β s , which varies γ only in e direction, i.e. β s is of the form β s ( t ) = ( t, θ s ( t ))For those variations we have already computed that if θ s is critical with respect to the L -action then β s is critical with respect to the Finsler length.We define sets V, V ⊂ ST via V = { ( x, v ) ∈ ST | x ∈ T , v > } , V = { (0 , h, v ) ∈ ST | h ∈ S , v > } From the completeness of the time-dependent Euler-Lagrange flow of L and proposition3.5 it follows that the lifts to R of geodesics c v with v > R · e ⊂ R and pass through the section V after finite time. Thus, the first returnmap R : V → V of the geodesic flow is well-defined. Proposition 3.6.
The return map R : V → V is conjugated via a diffeomorphism tothe twist map ˆ f .Proof. To see that the return map R is conjugated to ˆ f observe that R is given by R (0 , h, v , v ) = (cid:18) , θ (1) , (1 , θ ′ (1)) F (0 , θ (1) , , θ ′ (1)) (cid:19) (1)where θ : R → R is the Euler-Lagrange solution of the lifted Lagrangian L with θ (0) = h and θ ′ (0) = v v . This is true because after proposition 3.5 the curve γ : R → R with γ ( t ) = ( t, θ ( t )) is a reparametrized lift of the geodesic c : R → T with initial values c (0) = (0 , h ) and ˙ c (0) = ( v , v ). The reparametrized lift γ passes through a translate V + e of V again for the first time at time t = 1 and thus the return map R maps70 , h, v , v ) to ( γ (1) , ˙ γ (1) F ( γ (1) , ˙ γ (1) ), which is equal to the expression in equation (1). Thediffeomorphism g : V → Z conjugating R and ˆ f is given by g (0 , h, v , v ) = (cid:18) h, v v (cid:19) with inverse g − ( x, y ) = (cid:18) , x, (1 , y ) F (0 , x, , y ) (cid:19) Proposition 3.7.
The first return map R has positive metric entropy.Proof. Let g : V → Z be the diffeomorphism conjugating R to ˆ f . Let I ⊂ Z denote thestochastic island for ˆ f , i.e. every point x ∈ I has positive maximal Lyapunov exponentand Area( I ) = R I | dx ∧ dy | >
0. From the conjugacy of ˆ f and R it follows that themaximal Lyapunov exponent of every v ∈ g − ( I ) remains positive. Let ω denote thearea form of V obtained by restricting the differential dλ of the standard Liouville form λ to V . The return map R preserves ω . Since the pullback g ∗ ( dx ∧ dy ) and ω are botharea forms there is a positive function j : V → R > , such that j · ω = g ∗ ( dx ∧ dy )Since j is positive we have that R g − ( I ) ω = 0 if and only if R g − ( I ) j · ω = 0. And since Z g − ( I ) j · ω = Z g − ( I ) g ∗ ( dx ∧ dy ) = Z I dx ∧ dy we have that Area( g − ( I )) >
0. Consequently, R has positive metric entropy. References [1] P. Berger, D. Turaev,
On Herman’s Positive Entropy Conjecture , Advances in Math-ematics 349, 1234 - 1288 (2019)[2] M. Herman,
Sur les courbes invariantes par les diff´eomorphismes de l’anneau , vol. I,Aste´erisque, 103-104 (1983)[3] J. Moser,
Monotone twist mappings and the calculus of variations , Ergodic Theoryand Dynamical Systems, Vol. 6 (1986)[4] J. P. Schr¨oder,